Lesson Objectives At the end of the lesson

Lesson Objectives At the end of the lesson, students can: • Perform linear transformations on Random Variables. • Determine the mean and standard deviation of transformed Random Variables. • Determine the mean and standard deviation of sums and differences of Random Variables.

Warm-up--Pete’s Jeep Tours Number of Passengers (X) Probability 2. 15 3. 25 4. 35 5. 20 6. 05

Linear Transformation Review

Linear Transformation of Random Variables How are the probability distributions of random variables affected by similar transformations? Let’s follow an example through this…

Pete’s Jeep Tours—Effect of Mult by a constant Total collected (C) Probability

Pete’s Jeep Tours—Effect of Adding a constant Total collected (V) Probability

Rules for MEANS RULES FOR MEANS Means behave like averages! Rule 1: If X is a random variable and a and b are fixed numbers, then μa+bx = a + bμx μ a–bx = a – bμx Rule 2: If X and Y are random variables, then μ x + y = μx + μy μ x – y = μx – μy In other words, the mean of the sum = sum of the means and the mean of the difference = difference of the means.

Linear Transforms of Random Variables EXAMPLE: Consider the data on the distribution of the number of communications units sold by the military division and the civilian division. Military Division: # of Units Sold (X) Probability 1000. 1 3000. 3 300. 4 500. 5 5000 10, 000. 4. 2 Civilian Division: # of Units Sold (Y) Probability 750. 1 (a) Find the mean # of units sold collectively by both divisions. Let X = Let Y = μx = μy = μX+Y =

Linear Transforms of Random Variables EXAMPLE: Consider the data on the distribution of the number of communications units sold by the military division and the civilian division. (b) The company makes a profit of $2, 000 per military units sold and $3500 on each civilian unit sold. What will next year’s mean profit from military sales be? What will next year’s mean profit from civilian sales be? (c) Suppose we multiplied each value of X by 2 and added 10. (2 X + 10). How would this affect the mean?

Rules for Variances RULES FOR VARIANCE Rule 1: If X is a random variable and a and b are fixed numbers, then σ2 a+bx = b 2σ2 x σ2 a-bx = b 2σ2 x NOTE: Multiplying X by a constant “b” multiplies the variance of X by the “b 2”. The variance of X + a is the same as the variance of X. Rule 2: If X and Y are independent random variables, then σ2 x+y = σ2 x + σ2 y σ2 x-y = σ2 x + σ2 y In other words: ALWAYS ADD the variances, then take square root (This is called the “Addition rule for variances of independent random variables. ”)

Rules for Means and Variances Rules for Variance NOTE: When random variables are not independent, the variance of their sum depends on the relationship between them as well as on their individual variances. We use ρ (Greek letter “rho”) for the correlation between two random variables. The correlation ρ is a number between -1 and 1 that measures the strength and direction of the linear relationship between the two variables. The correlation between two independent random variables is zero. We will not be looking into detail with variance of not independent random variables.

Just remember. . . MORE VARIABLES MEANS MORE VARIABILITY!!! Always ADD variances only if the two random variables are independent!

Pete’s Sister Erin Number of Passengers (Y) Probability 2. 3 3. 4 4. 2 5. 1

Independent Random Variables If knowing whether any event involving “X alone has occurred” tells us nothing about the occurrence of any event involving “Y alone”, and vice versa, then X and Y are independent random variables. Probability models often assume independence when the random variables describe outcomes that appear unrelated to each other. You should always ask whether the assumption of independence seems reasonable! Are X, the number of Pete’s passengers on a random day, and Y, the number of Erin’s passengers on a random day, independent?

Pete’s and Erin’s Combined Business Number of Passengers (T) Probability 4 5 6 . 045. 135. 235 7. 265 8 9 . 190. 095 (See page 366 in book) 10 11 . 030 . 005

EXAMPLE: Earlier, we defined X = the number of passengers on Pete’s trip, Y = the number of passengers on Erin’s trip, and C = the amount of money that Pete collects on a randomly selected day. We also found the means and standard deviations of these variables: μx = 3. 75 μy = 3. 10 μC = 562. 50 σx = 1. 090 σy = 0. 943 σC = 163. 50 (a) Erin charges $175 per passenger for her trip. Let G = the amount of money that she collects on a randomly selected day. Find the mean and standard deviation of G.

EXAMPLE: Earlier, we defined X = the number of passengers on Pete’s trip, Y = the number of passengers on Erin’s trip, and C = the amount of money that Pete collects on a randomly selected day. We also found the means and standard deviations of these variables: μx = 3. 75 μy = 3. 10 μC = 562. 50 σx = 1. 090 σy = 0. 943 σC = 163. 50 (b) Calculate the mean and the standard deviation of the total amount that Pete and Erin collect on a randomly chosen day.

Difference of Random Variables •

Lesson Objectives At the end of the lesson, students can: • Perform linear transformations on Random Variables. • Determine the mean and standard deviation of transformed Random Variables. • Determine the mean and standard deviation of sums and differences of Random Variables.

HW • Section 6. 2 exercises: • P. 378/ #36, 37, 39 -41, 43